The solution algorithm of CoMLSim approach is shown in Fig. REF  and described in detail in Alg. REF . Similar to traditional solvers, the CoMLSim approach discretizes the computational domain into smaller subdomains contain \(n^p\)  computational elements, where \(p\)  refers to spatial dimensionality and \(n\)  is predetermined. Each subdomain has a constant physical size and represents both PDE solutions and conditions. For example, a subdomain cutting across the cylinder in Fig. REF  represents the geometry and boundary conditions of the part-cylinder as well as the corresponding solution. In a steady-state problem, the solutions on the domain are initialized either with uniform solutions or they are generated from coarse-grid PDE solvers. The initial solution \(\vec{p}=(u(\vec{x}), v(\vec{x}))\)  is shown with randomly oriented flow vectors in Fig. REF . Pretrained encoders \(e_{\vec{p}}, e_g, e_b, e_s\)  are used to encode the initial solutions as well as user-specified PDE conditions into lower-dimensional latent vectors, \(\eta _{\vec{p}}\)  corresponding to PDE solutions and \(\eta _g, \eta _b, \eta _s\)  corresponding to geometry, boundary conditions and source terms, respectively. Solution and condition latent vectors on groups of neighboring subdomains are concatenated together evaluated using a pre-trained flux conservation autoencoder (\(\Theta \) ) to get a new set of solution latent vectors (\(\eta ^{\prime }_{\vec{p}}\) ) on each subdomain, which are more locally consistent than the original vectors. The new solution latent vectors are iteratively passed through \(\Theta \)  to improve their local consistency and the iteration is stopped when the \(L_2\)  norm of change in solution latent vectors meets a specified tolerance, otherwise the iteration continues with the updated latent vectors. The latent vectors of the PDE conditions are not updated and help in steering the solution latent vectors to an equilibrium state that is decoded to PDE solutions using pretrained decoders \((g_{\vec{p}})\)  on the computational domain. The converged solution in Fig. REF  is represented with flow vectors that are locally consistent with neighboring subdomains.The iterative procedure used in the CoMLSim approach can be implemented using several linear or non-linear equation solvers, such as Fixed point iterations [1]}, Gauss Seidel, Newton's method etc., that are used in commercial PDE solvers. Out of those we have explored Point Jacobi and Gauss Seidel, which are described below. The use of autoencoders for compressed solution and condition representation in tandem with iterative inferencing solution algorithm is inspired from [2]}, [3]} but the main difference in this work is that the solution procedure is carried out on local subdomains as opposed to entire computational domains to align with principles used in traditional PDE solvers. Supplementary material section G explains our algorithm in more details using an example.
